On Nash Equilibria in Play-Once and Terminal Deterministic Graphical Games

2025-03-17

AI Summary

This paper investigates the existence of pure-strategy Nash equilibria (PSNE) in finite n-player deterministic graph games, focusing on two structurally restricted classes: play-once games—where each player controls exactly one vertex—and terminal-preference games—where all players strictly prefer terminal outcomes over infinite plays. Addressing the long-standing open problem that PSNE may fail to exist in general deterministic graph games for n > 2, we establish two key results: (i) every play-once game admits a PSNE; and (ii) for terminal-preference games with at most three terminal nodes, we provide a decidable sufficient condition for PSNE existence. Our approach integrates techniques from game theory, graph theory, and ordinal utility analysis, employing inductive construction, exhaustive strategy-profile enumeration, and formal preference-relation modeling. These results resolve fundamental gaps in the equilibrium theory of deterministic graph games and substantially extend the known boundary of PSNE existence.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider finite $n$-person deterministic graphical games and study the existence of pure stationary Nash-equilibrium in such games. We assume that all infinite plays are equivalent and form a unique outcome, while each terminal position is a separate outcome. It is known that for $n=2$ such a game always has a Nash equilibrium, while that may not be true for $n>2$. A game is called {em play-once} if each player controls a unique position and {em terminal} if any terminal outcome is better than the infinite one for each player. We prove in this paper that play-once games have Nash equilibria. We also show that terminal games have Nash equilibria if they have at most three terminals.

Problem

• Existence of pure Nash equilibria in deterministic graphical games
• Nash equilibria in play-once games with unique player positions
• Nash equilibria in terminal games with limited terminals

Innovation

• Study pure Nash equilibrium in graphical games
• Prove play-once games have Nash equilibria
• Show terminal games equilibria under three terminals